Let $p: E \rightarrow B$ a fiber fundle with fiber $F$. I'd like to prove that if $U$ is a good-cover of $B$ then the Euler characteristic, that I denote with $\chi$ is $\chi(E)= \sum_{p,q}\sum_{\alpha_{0}, \cdots, \alpha_{p}} (-1)^{p+q}\dim H^{q}(p^{-1}(U_{\alpha_{0} \le \cdots le \alpha_{p}}))$. Then deduces from this fact that $\chi(E)=\chi(F)\chi(B)$.
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1Are you familiar with the Mayer-Vietoris spectral sequence for a covering? – Mariano Suárez-Álvarez Jan 17 '13 at 06:58
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Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. – Julian Kuelshammer Jan 17 '13 at 09:01
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No a lot... but if you are accurate I can try to understand... – ArthurStuart Jan 17 '13 at 08:02
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@ArthurStuart: I've merged your two accounts. You should be able to log-in with either set of credentials to the same account now. This way you can comment on and edit your own question. – Willie Wong Jan 17 '13 at 09:06
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@ Mariano Suárez-Alvarez Could I find a simple proof of my claim somewhere? – ArthurStuart Jan 17 '13 at 09:20
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To amplify on Mariano's comment: If $\{U_i\}$ is an open cover of $X$, then there is a convergent Mayer-Vietoris spectral sequence $$ \bigoplus H^p(U_{i_1}\cap \cdots \cap U_{i_q}) \implies H^{p+q}(X).$$ Now use that the Euler characteristic of a spectral sequence, i.e. $\sum_{p,q} (-1)^{p+q}\dim E_{r}^{p,q}$, does not depend on $r$. This proves the first part.
If you don't know spectral sequences then I think you can also do this by the usual Mayer-Vietoris sequence for a cover with two open sets, and induction over the number of opens in your good cover. But I suspect this could be messier combinatorially.
For the second use that $p^{-1}(U_{i_1}\cap \cdots \cap U_{i_q}) \cong (U_{i_1}\cap \cdots \cap U_{i_q}) \times F$ and the Künneth theorem.
Dan Petersen
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What is $X$? I haven't understood how is made the Mayer-Vietoris spectral sequece – ArthurStuart Jan 17 '13 at 21:18
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Where can I find a clean general formulation of the (co)homological MV spectral sequence? Weibel or McCleary doesn't have it. I'm guessing it goes like this: for any subsets $(U_i){i\in I}$ of a topological space $X$ such that their interiors cover $X$, there is a homological spectral sequence of $R$-modules with second page $E^2{p,q}=???$ and converging to $H_{p+q}(X;R)$, and a cohomological spectral sequence of $R$-algebras with second page $E_2^{p,q}=???$ and converging to $H^{p+q}(X;R)$. Is this also true for other topological (co)homology theories? Does $I$ need to be countable? – Leo Jan 09 '14 at 01:24
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Maybe $E_{p,q}^2=\bigoplus_{I'\subseteq I, |I'|=p}H_q(\bigcap_{i\in I'}U_i)$? Shouldn't there be unions of $U_i$ somewhere? – Leo Jan 09 '14 at 01:28